Difference between revisions of "2007 AMC 8 Problems/Problem 16"
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== Solution == | == Solution == | ||
| − | The circumference of a circle is obtained by simply multiplying the radius by | + | The circumference of a circle is obtained by simply multiplying the radius by 2 pi. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by Pi. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is A |
==Video Solution by WhyMath== | ==Video Solution by WhyMath== | ||
Revision as of 18:26, 1 August 2021
Problem
Amanda Reckonwith draws five circles with radii 
and 
. Then for each circle she plots the point 
,
where 
is its circumference and 
is its area. Which of the
following could be her graph?
![[asy] size(75); pair A= (1.5,2) , B= (3,4) , C= (4.5,7) , D= (6,11) , E= (7.5,16) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]](http://latex.artofproblemsolving.com/5/4/4/5441a586e3c38f07d4057cb9867265ca14fdeb6e.png)
![[asy] size(75); pair A= (1.5,9) , B= (3,6) , C= (4.5,6) , D= (6,9) , E= (7.5,15) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]](http://latex.artofproblemsolving.com/2/3/0/230f178f3cd06089682b3fe4a6a9ecf9580ef492.png)
![[asy] size(75); pair A= (1.5,2) , B= (3,6) , C= (4.5,8) , D= (6,6) , E= (7.5,2) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]](http://latex.artofproblemsolving.com/9/5/d/95defaa77d8b1dd10283c438752810174f3e6d94.png)
![[asy] size(75); pair A= (1.5,2) , B= (3,5) , C= (4.5,8) , D= (6,11) , E= (7.5,14) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]](http://latex.artofproblemsolving.com/9/d/4/9d43329cbcfb1249b7fcf1fc15cd2b90bc666ef4.png)
![[asy] size(75); pair A= (1.5,15) , B= (3,10) , C= (4.5,6) , D= (6,3) , E= (7.5,1) ; draw((0,-1)--(0,16)); draw((-1,0)--(16,0)); dot(A^^B^^C^^D^^E); label("$A$", (0,8), W); label("$C$", (8,0), S);[/asy]](http://latex.artofproblemsolving.com/5/7/d/57d2b3b4be42360bf68db966abbc090187b5cdb3.png)
Solution
The circumference of a circle is obtained by simply multiplying the radius by 2 pi. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by Pi. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is A
Video Solution by WhyMath
~savannahsolver
See Also
| 2007 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
